What MSNBC carefully failed to mention
Is that those UNHCR numbers also include the Iraqis who fled or were internally displaced during Saddam's regime. Of course, mentioning that would spoil the "Saddam's Peaceful Paradise" meme.
I don't know how many were pre-war. The UNHCR article only calls it "a sizable proportion."
I do know that during the 1991 uprising, there were 1.2 million refugees in Iran alone, but I also know that a number of these returned to Iraq in later years. Exact numbers are hard to come by.